Optimal. Leaf size=849 \[ \frac {x \sqrt {d+e x}}{4 a \left (a+c x^2\right )^2}+\frac {\sqrt {d+e x} \left (a d e+\left (6 c d^2+5 a e^2\right ) x\right )}{16 a^2 \left (c d^2+a e^2\right ) \left (a+c x^2\right )}+\frac {e \left (6 c^{3/2} d^3+8 a \sqrt {c} d e^2+\sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{32 \sqrt {2} a^2 c^{3/4} \left (c d^2+a e^2\right )^{3/2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {e \left (6 c^{3/2} d^3+8 a \sqrt {c} d e^2+\sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{32 \sqrt {2} a^2 c^{3/4} \left (c d^2+a e^2\right )^{3/2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {e \left (6 c^{3/2} d^3+8 a \sqrt {c} d e^2-\sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \log \left (\sqrt {c d^2+a e^2}-\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{64 \sqrt {2} a^2 c^{3/4} \left (c d^2+a e^2\right )^{3/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {e \left (6 c^{3/2} d^3+8 a \sqrt {c} d e^2-\sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \log \left (\sqrt {c d^2+a e^2}+\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{64 \sqrt {2} a^2 c^{3/4} \left (c d^2+a e^2\right )^{3/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}} \]
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Rubi [A]
time = 1.89, antiderivative size = 849, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 8, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {751, 837, 841,
1183, 648, 632, 212, 642} \begin {gather*} \frac {\sqrt {d+e x} x}{4 a \left (c x^2+a\right )^2}+\frac {e \left (6 c^{3/2} d^3+8 a \sqrt {c} e^2 d+\sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{32 \sqrt {2} a^2 c^{3/4} \left (c d^2+a e^2\right )^{3/2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {e \left (6 c^{3/2} d^3+8 a \sqrt {c} e^2 d+\sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{32 \sqrt {2} a^2 c^{3/4} \left (c d^2+a e^2\right )^{3/2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {e \left (6 c^{3/2} d^3+8 a \sqrt {c} e^2 d-\sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \log \left (\sqrt {c} (d+e x)-\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c d^2+a e^2}\right )}{64 \sqrt {2} a^2 c^{3/4} \left (c d^2+a e^2\right )^{3/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {e \left (6 c^{3/2} d^3+8 a \sqrt {c} e^2 d-\sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \log \left (\sqrt {c} (d+e x)+\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c d^2+a e^2}\right )}{64 \sqrt {2} a^2 c^{3/4} \left (c d^2+a e^2\right )^{3/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\sqrt {d+e x} \left (a d e+\left (6 c d^2+5 a e^2\right ) x\right )}{16 a^2 \left (c d^2+a e^2\right ) \left (c x^2+a\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
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Rule 212
Rule 632
Rule 642
Rule 648
Rule 751
Rule 837
Rule 841
Rule 1183
Rubi steps
\begin {align*} \int \frac {\sqrt {d+e x}}{\left (a+c x^2\right )^3} \, dx &=\frac {x \sqrt {d+e x}}{4 a \left (a+c x^2\right )^2}-\frac {\int \frac {-3 d-\frac {5 e x}{2}}{\sqrt {d+e x} \left (a+c x^2\right )^2} \, dx}{4 a}\\ &=\frac {x \sqrt {d+e x}}{4 a \left (a+c x^2\right )^2}+\frac {\sqrt {d+e x} \left (a d e+\left (6 c d^2+5 a e^2\right ) x\right )}{16 a^2 \left (c d^2+a e^2\right ) \left (a+c x^2\right )}+\frac {\int \frac {\frac {1}{4} c d \left (12 c d^2+13 a e^2\right )+\frac {1}{4} c e \left (6 c d^2+5 a e^2\right ) x}{\sqrt {d+e x} \left (a+c x^2\right )} \, dx}{8 a^2 c \left (c d^2+a e^2\right )}\\ &=\frac {x \sqrt {d+e x}}{4 a \left (a+c x^2\right )^2}+\frac {\sqrt {d+e x} \left (a d e+\left (6 c d^2+5 a e^2\right ) x\right )}{16 a^2 \left (c d^2+a e^2\right ) \left (a+c x^2\right )}+\frac {\text {Subst}\left (\int \frac {-\frac {1}{4} c d e \left (6 c d^2+5 a e^2\right )+\frac {1}{4} c d e \left (12 c d^2+13 a e^2\right )+\frac {1}{4} c e \left (6 c d^2+5 a e^2\right ) x^2}{c d^2+a e^2-2 c d x^2+c x^4} \, dx,x,\sqrt {d+e x}\right )}{4 a^2 c \left (c d^2+a e^2\right )}\\ &=\frac {x \sqrt {d+e x}}{4 a \left (a+c x^2\right )^2}+\frac {\sqrt {d+e x} \left (a d e+\left (6 c d^2+5 a e^2\right ) x\right )}{16 a^2 \left (c d^2+a e^2\right ) \left (a+c x^2\right )}+\frac {\text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (-\frac {1}{4} c d e \left (6 c d^2+5 a e^2\right )+\frac {1}{4} c d e \left (12 c d^2+13 a e^2\right )\right )}{\sqrt [4]{c}}-\left (-\frac {1}{4} c d e \left (6 c d^2+5 a e^2\right )-\frac {1}{4} \sqrt {c} e \sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right )+\frac {1}{4} c d e \left (12 c d^2+13 a e^2\right )\right ) x}{\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt {d+e x}\right )}{8 \sqrt {2} a^2 c^{5/4} \left (c d^2+a e^2\right )^{3/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (-\frac {1}{4} c d e \left (6 c d^2+5 a e^2\right )+\frac {1}{4} c d e \left (12 c d^2+13 a e^2\right )\right )}{\sqrt [4]{c}}+\left (-\frac {1}{4} c d e \left (6 c d^2+5 a e^2\right )-\frac {1}{4} \sqrt {c} e \sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right )+\frac {1}{4} c d e \left (12 c d^2+13 a e^2\right )\right ) x}{\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt {d+e x}\right )}{8 \sqrt {2} a^2 c^{5/4} \left (c d^2+a e^2\right )^{3/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}\\ &=\frac {x \sqrt {d+e x}}{4 a \left (a+c x^2\right )^2}+\frac {\sqrt {d+e x} \left (a d e+\left (6 c d^2+5 a e^2\right ) x\right )}{16 a^2 \left (c d^2+a e^2\right ) \left (a+c x^2\right )}-\frac {\left (e \left (6 c^{3/2} d^3+8 a \sqrt {c} d e^2-\sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right )\right ) \text {Subst}\left (\int \frac {-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}+2 x}{\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt {d+e x}\right )}{64 \sqrt {2} a^2 c^{3/4} \left (c d^2+a e^2\right )^{3/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\left (e \left (6 c^{3/2} d^3+8 a \sqrt {c} d e^2-\sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right )\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}+2 x}{\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt {d+e x}\right )}{64 \sqrt {2} a^2 c^{3/4} \left (c d^2+a e^2\right )^{3/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\left (e \left (6 c^{3/2} d^3+8 a \sqrt {c} d e^2+\sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right )\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt {d+e x}\right )}{64 a^2 c \left (c d^2+a e^2\right )^{3/2}}+\frac {\left (e \left (6 c^{3/2} d^3+8 a \sqrt {c} d e^2+\sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right )\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt {d+e x}\right )}{64 a^2 c \left (c d^2+a e^2\right )^{3/2}}\\ &=\frac {x \sqrt {d+e x}}{4 a \left (a+c x^2\right )^2}+\frac {\sqrt {d+e x} \left (a d e+\left (6 c d^2+5 a e^2\right ) x\right )}{16 a^2 \left (c d^2+a e^2\right ) \left (a+c x^2\right )}-\frac {e \left (6 c^{3/2} d^3+8 a \sqrt {c} d e^2-\sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \log \left (\sqrt {c d^2+a e^2}-\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{64 \sqrt {2} a^2 c^{3/4} \left (c d^2+a e^2\right )^{3/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {e \left (6 c^{3/2} d^3+8 a \sqrt {c} d e^2-\sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \log \left (\sqrt {c d^2+a e^2}+\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{64 \sqrt {2} a^2 c^{3/4} \left (c d^2+a e^2\right )^{3/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}-\frac {\left (e \left (6 c^{3/2} d^3+8 a \sqrt {c} d e^2+\sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right )\right ) \text {Subst}\left (\int \frac {1}{2 \left (d-\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}\right )-x^2} \, dx,x,-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}+2 \sqrt {d+e x}\right )}{32 a^2 c \left (c d^2+a e^2\right )^{3/2}}-\frac {\left (e \left (6 c^{3/2} d^3+8 a \sqrt {c} d e^2+\sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right )\right ) \text {Subst}\left (\int \frac {1}{2 \left (d-\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}\right )-x^2} \, dx,x,\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}+2 \sqrt {d+e x}\right )}{32 a^2 c \left (c d^2+a e^2\right )^{3/2}}\\ &=\frac {x \sqrt {d+e x}}{4 a \left (a+c x^2\right )^2}+\frac {\sqrt {d+e x} \left (a d e+\left (6 c d^2+5 a e^2\right ) x\right )}{16 a^2 \left (c d^2+a e^2\right ) \left (a+c x^2\right )}+\frac {e \left (6 c^{3/2} d^3+8 a \sqrt {c} d e^2+\sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}-\sqrt {2} \sqrt {d+e x}\right )}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{32 \sqrt {2} a^2 c^{3/4} \left (c d^2+a e^2\right )^{3/2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {e \left (6 c^{3/2} d^3+8 a \sqrt {c} d e^2+\sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}+\sqrt {2} \sqrt {d+e x}\right )}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{32 \sqrt {2} a^2 c^{3/4} \left (c d^2+a e^2\right )^{3/2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {e \left (6 c^{3/2} d^3+8 a \sqrt {c} d e^2-\sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \log \left (\sqrt {c d^2+a e^2}-\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{64 \sqrt {2} a^2 c^{3/4} \left (c d^2+a e^2\right )^{3/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {e \left (6 c^{3/2} d^3+8 a \sqrt {c} d e^2-\sqrt {c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \log \left (\sqrt {c d^2+a e^2}+\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{64 \sqrt {2} a^2 c^{3/4} \left (c d^2+a e^2\right )^{3/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 1.14, size = 367, normalized size = 0.43 \begin {gather*} \frac {\frac {2 \sqrt {a} \sqrt {d+e x} \left (6 c^2 d^2 x^3+a^2 e (d+9 e x)+a c x \left (10 d^2+d e x+5 e^2 x^2\right )\right )}{\left (c d^2+a e^2\right ) \left (a+c x^2\right )^2}+\frac {i \left (12 c d^2+18 i \sqrt {a} \sqrt {c} d e-5 a e^2\right ) \tan ^{-1}\left (\frac {\sqrt {-c d-i \sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d+i \sqrt {a} e}\right )}{\sqrt {c} \left (\sqrt {c} d+i \sqrt {a} e\right ) \sqrt {-c d-i \sqrt {a} \sqrt {c} e}}-\frac {i \left (12 c d^2-18 i \sqrt {a} \sqrt {c} d e-5 a e^2\right ) \tan ^{-1}\left (\frac {\sqrt {-c d+i \sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d-i \sqrt {a} e}\right )}{\sqrt {c} \left (\sqrt {c} d-i \sqrt {a} e\right ) \sqrt {-c d+i \sqrt {a} \sqrt {c} e}}}{32 a^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {e x +d}}{\left (c \,x^{2}+a \right )^{3}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 3532 vs.
\(2 (678) = 1356\).
time = 3.07, size = 3532, normalized size = 4.16 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.19, size = 1056, normalized size = 1.24 \begin {gather*} -\frac {{\left ({\left (a^{2} c d^{2} e + a^{3} e^{3}\right )}^{2} {\left (6 \, c d^{2} e + 5 \, a e^{3}\right )} {\left | c \right |} - 2 \, {\left (3 \, \sqrt {-a c} a c^{2} d^{5} e + 7 \, \sqrt {-a c} a^{2} c d^{3} e^{3} + 4 \, \sqrt {-a c} a^{3} d e^{5}\right )} {\left | -a^{2} c d^{2} e - a^{3} e^{3} \right |} {\left | c \right |} + {\left (12 \, a^{3} c^{4} d^{8} e + 37 \, a^{4} c^{3} d^{6} e^{3} + 38 \, a^{5} c^{2} d^{4} e^{5} + 13 \, a^{6} c d^{2} e^{7}\right )} {\left | c \right |}\right )} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-\frac {a^{2} c^{2} d^{3} + a^{3} c d e^{2} + \sqrt {{\left (a^{2} c^{2} d^{3} + a^{3} c d e^{2}\right )}^{2} - {\left (a^{2} c^{2} d^{4} + 2 \, a^{3} c d^{2} e^{2} + a^{4} e^{4}\right )} {\left (a^{2} c^{2} d^{2} + a^{3} c e^{2}\right )}}}{a^{2} c^{2} d^{2} + a^{3} c e^{2}}}}\right )}{32 \, {\left (a^{4} c^{3} d^{4} e - \sqrt {-a c} a^{3} c^{3} d^{5} - 2 \, \sqrt {-a c} a^{4} c^{2} d^{3} e^{2} + 2 \, a^{5} c^{2} d^{2} e^{3} - \sqrt {-a c} a^{5} c d e^{4} + a^{6} c e^{5}\right )} \sqrt {-c^{2} d + \sqrt {-a c} c e} {\left | -a^{2} c d^{2} e - a^{3} e^{3} \right |}} - \frac {{\left ({\left (a^{2} c d^{2} e + a^{3} e^{3}\right )}^{2} {\left (6 \, c d^{2} e + 5 \, a e^{3}\right )} {\left | c \right |} + 2 \, {\left (3 \, \sqrt {-a c} a c^{2} d^{5} e + 7 \, \sqrt {-a c} a^{2} c d^{3} e^{3} + 4 \, \sqrt {-a c} a^{3} d e^{5}\right )} {\left | -a^{2} c d^{2} e - a^{3} e^{3} \right |} {\left | c \right |} + {\left (12 \, a^{3} c^{4} d^{8} e + 37 \, a^{4} c^{3} d^{6} e^{3} + 38 \, a^{5} c^{2} d^{4} e^{5} + 13 \, a^{6} c d^{2} e^{7}\right )} {\left | c \right |}\right )} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-\frac {a^{2} c^{2} d^{3} + a^{3} c d e^{2} - \sqrt {{\left (a^{2} c^{2} d^{3} + a^{3} c d e^{2}\right )}^{2} - {\left (a^{2} c^{2} d^{4} + 2 \, a^{3} c d^{2} e^{2} + a^{4} e^{4}\right )} {\left (a^{2} c^{2} d^{2} + a^{3} c e^{2}\right )}}}{a^{2} c^{2} d^{2} + a^{3} c e^{2}}}}\right )}{32 \, {\left (a^{4} c^{3} d^{4} e + \sqrt {-a c} a^{3} c^{3} d^{5} + 2 \, \sqrt {-a c} a^{4} c^{2} d^{3} e^{2} + 2 \, a^{5} c^{2} d^{2} e^{3} + \sqrt {-a c} a^{5} c d e^{4} + a^{6} c e^{5}\right )} \sqrt {-c^{2} d - \sqrt {-a c} c e} {\left | -a^{2} c d^{2} e - a^{3} e^{3} \right |}} + \frac {6 \, {\left (x e + d\right )}^{\frac {7}{2}} c^{2} d^{2} e - 18 \, {\left (x e + d\right )}^{\frac {5}{2}} c^{2} d^{3} e + 18 \, {\left (x e + d\right )}^{\frac {3}{2}} c^{2} d^{4} e - 6 \, \sqrt {x e + d} c^{2} d^{5} e + 5 \, {\left (x e + d\right )}^{\frac {7}{2}} a c e^{3} - 14 \, {\left (x e + d\right )}^{\frac {5}{2}} a c d e^{3} + 23 \, {\left (x e + d\right )}^{\frac {3}{2}} a c d^{2} e^{3} - 14 \, \sqrt {x e + d} a c d^{3} e^{3} + 9 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{2} e^{5} - 8 \, \sqrt {x e + d} a^{2} d e^{5}}{16 \, {\left (a^{2} c d^{2} + a^{3} e^{2}\right )} {\left ({\left (x e + d\right )}^{2} c - 2 \, {\left (x e + d\right )} c d + c d^{2} + a e^{2}\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.22, size = 2500, normalized size = 2.94 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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